Admissible wavelets on the Siegel domain of type one

LetSp(n, R) be the sympletic group, and letK _n * be its maximal compact subgroup. ThenG=Sp(n,R)/K _n * can be realized as the Siegel domain of type one. The square-integrable representation ofG gives the admissible wavelets AW and wavelet transform. The characterization of admissibility condition in terms of the Fourier transform is given. The Bergman kernel follows from the viewpoint of coherent state. With the Laguerre polynomials, Hermite polynomials and Jacobi polynomials, two kinds of orthogonal bases for AW are given, and they then give orthogonal decompositions ofL ²-space on the Siegel domain of type one ℒ(ℋ _n , |y| *dxdy). Project supported in part by the National Natural Science Foundation of China (Grant No. 19631080).     

LP-boundedness of multilinear oscillatory singular integralswith Calderon-Zygmund kernel

LP-boundedness is obtained for a class of multilinear oscillatory singular integrals with CalderonZygmund kernel.     

On Dissipative Phenomena of the Interaction of Hamiltonian Systems

The dynamics is under study of a composite Hamiltonian system that is the union of a finite-dimensional nonlinear system and an infinite-dimensional linear system with quadratic interaction Hamiltonian. The dynamics of the finite-dimensional subsystem is determined by a nonlinear integro-differential equation with a relaxation kernel. We prove existence and uniqueness theorems and find a priori estimates for a solution. Under some assumptions on the form of interaction, the solution to the finite-dimensional subsystem converges to one of the critical points of the effective Hamiltonian.     

On the optimal approximation of bounded linear functionals in Hilbert spaces with inner product invariant in rotation or translation

Optimal numerical approximation of bounded linear functionals by weighted sums in Hilbert spaces of functions defined in a domain B ? $\text{C}$ or B ? $\text{R}$^m, invariant in rotation or translation (e.g. circle, circular annulus, ball, spherical shell, strip of the complex plane) and equipped with inner product invariant in rotation or translation are considered. The weights and error functional norms for optimal approximate rules based on nodes located angle-equidistant on concentric spheres or circles of B, for B invariant in rotation, and on nodes located equispaced on in B lying line, for B invariant in translation, are explicitly given in terms of the kernel function of the Hilbert space. A number of concrete Hilbert spaces satisfying the required conditions are listed.     

Diffusion processes on complete riemannian manifolds

ＤＩＦＦＵＳＩＯＮＰＲＯＣＥＳＳＥＳＯＮＣＯＭＰＬＥＴＥＲＩＥＭＡＮＮＩＡＮＭＡＮＩＦＯＬＤＳＱＩＡＮＺＨＯＮＧＭＩＮ（钱忠民）（ＤｅｐａｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ,ＳｈａｎｇｈａｉＩｎｓｔｉｔｕｔｅｏｆＲａｉｌｗａｙＴｅｃ...     

On a convolution inequality of Saitoh

Let be in the class of locally integrable functions on . Define the convolution product inductively by and for . The inequality

is obtained for each , . Further, the constant is shown to be the best possible, and the nonzero extremal functions are determined.

     

Average sampling and reconstruction for reproducing kernel stochastic signals

This paper mainly considers the problem of reconstructing a reproducing kernel stochastic signal from its average samples. First, a uniform convergence result for reconstructing the deterministic reproducing kernel signals by an iterative algorithm is established. Then, we prove that the quadratic sum of the corresponding reconstructed functions is uniformly bounded. Moreover, the reconstructed functions provide a frame expansion in the special case p = 2. Finally, the mean square convergence for recovering a weighted reproducing kernel stochastic signal from its average samples is given under some decay condition for the autocorrelation function, which can be removed for the case p = 2. Copyright © 2015 John Wiley & Sons, Ltd.     

Reinforced Angle-Based Multicategory Support Vector Machines

The support vector machine (SVM) is a very popular classification tool with many successful applications. It was originally designed for binary problems with desirable theoretical properties. Although there exist various multicategory SVM (MSVM) extensions in the literature, some challenges remain. In particular, most existing MSVMs make use of k classification functions for a k-class problem, and the corresponding optimization problems are typically handled by existing quadratic programming solvers. In this article, we propose a new group of MSVMs, namely, the reinforced angle-based MSVMs (RAMSVMs), using an angle-based prediction rule with k ? 1 functions directly. We prove that RAMSVMs can enjoy Fisher consistency. Moreover, we show that the RAMSVM can be implemented using the very efficient coordinate descent algorithm on its dual problem. Numerical experiments demonstrate that our method is highly competitive in terms of computational speed, as well as classification prediction performance. Supplemental materials for the article are available online.     

Rapid Solution of Minimal Riesz Energy Problems

In , , we compute the solution to both the unconstrained and constrained Gauss variational problem, considered for the Riesz kernel of order and a pair of compact, disjoint, boundaryless ‐dimensional ‐manifolds , , where , each being charged with Borel measures with the sign prescribed. Such variational problems over a cone of Borel measures can be formulated as minimization problems over the corresponding cone of surface distributions belonging to the Sobolev–Slobodetski space , where and (see Harbrecht et al., Math. Nachr. 287 (2014), 48–69). We thus approximate the sought density by piecewise constant boundary elements and apply the primal‐dual active set strategy to impose the desired inequality constraints. The boundary integral operator which is defined by the Riesz kernel under consideration is efficiently approximated by means of an ‐matrix approximation. This particularly enables the application of a preconditioner for the iterative solution of the first‐order optimality system. Numerical results in are given to demonstrate our approach. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1535–1552, 2016